Fuzzy Bisimulations for Nondeterministic Fuzzy Transition Systems

Bisimulations are established forms of behavioral equivalences for discrete event systems. Recently, bisimulations have been introduced to fuzzy transition systems (FTSs) and developed quickly. Relational lifting by using weight functions is one of the most used methods in bisimulation research, which lifts a crisp relation on the state spaces to a crisp relation on the distributions of state spaces of FTSs. This article proposes a method, called fuzzy relational lifting method, via fuzzy similarity measures induced by residua (implications) in complete residuated lattices. In contrast with the existing method, fuzzy relational lifting method relaxes the conditions of lifting relation by using weight functions and is a generalization of the method. Some properties of fuzzy lifting operation are presented. Based on the fuzzy lifting operation, a fuzzy bisimulation for nondeterministic fuzzy transition systems is introduced, which is much more natural and robust than some behavioral measures, and can measure the degree of similarity between any two states, which are deemed to be equivalent intuitively but simply distinguished by other behavioral measures. Then some properties of fuzzy bisimulations are provided. Moreover, a fixed-point characterization of fuzzy bisimulations is given and the greatest fuzzy bisimulation is computed. Finally, a real-valued logic is introduced to characterize fuzzy bisimulations over Gödel algebra.